Analysis of correlations between time series with the use of the phase spectrum

In connection with the problem of revealing cause-effect relations among different climatic characteristics, methods for determining the dependence between time series on the basis of selected auto-and cross-covariance functions and periodograms expressed in terms of the Fourier coefficients of the initial series are discussed. Real time series do not always satisfy the conditions of statistical stationarity, so that their analysis requires a combination of statistical and deterministic approaches. The possibility of using the above numerical characteristics in relation to such series is considered. Characteristic features of describing finite segments of time series with the use of their Fourier coefficients is studied in detail. The main emphasis is on the determination of the time shifts (delays) at which the covariance between the series is maximal. The problems that appear during a practical implementation of the periodogram method for shift estimation are discussed. A previously unknown formula that is necessary for further studies and relates the Fourier transform of a selected correlation function to the periodogram of the series is derived.     

Lidar sounding of the optical parameter of atmospheric turbulence

The operation of a lidar intended for clear air turbulence (CAT) positioning on the basis of the backscatter enhancement (BSE) effect is analyzed using a turbulence model with a power-law spectrum. Systematic distortions occurring due to a need to regularize the lidar positioning problem solution are estimated. It is shown that the effect of molecular viscosity of air on the positioning result can be neglected if the wave parameter, which characterizes the diffraction manifestation, is higher than 3. This corresponds to sounding ranges of more than 1 km for optical or UV lidars. The analysis results show that the BSE lidar positioning accuracy weakly depends on the exponent in the turbulence spectrum in regions of severe turbulence. The results can justify a physical experiment for the design of an aircraft system for the lidar detection of CAT regions ahead of the flight course.     

Statistics of Air Temperature Spatial Variability in the Atmospheric Surface Layer

Surface-layer convection is investigated by analyzing multi-point measurements of temperature and velocity fluctuations at different sets of spatial points.The visual analysis of temperature and velocity fluctuations measured by sensors mounted on a mast of 36-m height clearly reveals the presence of large-scale convective cells (known as ramp structures) making large contributions to the heat transfer from the ground to lower atmosphere. The vertical temperature variability is described with the aid of empirical orthogonal functions derived from temperature covariance matrices for the heights of 1, 2, 5, 10, 18 and 36 m. Temporal-spatial correlation functions obtained allow estimates of a characteristic velocity scale, which may be interpreted as the downwind velocity of ramp structures.     

Akiva Moiseevich Yaglom (on his 85th birthday)

Personalia

Akiva Moiseevich Yaglom (on his 85th birthday)     

Random Processes with Stationary Increments and Composite Spectra

Izvestiya, Atmospheric and Oceanic Physics - A technique for deriving and interpreting fractal geophysical processes with power-law spectra of a different nature is described. Examples include the...     

Determining confidence intervals in analyzing climatic series

In the context of the problem of global warming, the question of how much the observed increase in the mean air temperature over the last decades is likely due only to natural variability is studied. It is assumed that an observed temperature-data series of length N is a segment of a stationary random sequence with a spectrum known on an interval that does not include the lowest frequencies. The variance σ²(N) of the sample mean m* and the standard deviation σ(N), which determines the width of the confidence interval of m*, are calculated for different continuations of the spectrum to the low-frequency region. For a continuation of the spectrum that boundlessly increases, as does ω^−2α (0 < α < 1/2) when the frequency ω tends to zero (red-noise spectrum), it is shown that, the closer the parameter α is to 1/2, the more slowly σ(N) tends to zero at N → ∞. Using an empirical spectrum of a global mean annual temperature series as an example, it is shown that the standard deviation significantly depends on the form of the spectrum in the lowest frequency region. An attempt has been made to assess the measure of standard-deviation uncertainty due to a lack of exact information on the spectrum in the low-frequency region. Important series characteristics—the equivalent number of independent observations N _eq(N) and the time scale of correlation T ₁(N) which are determined through the variance σ²(N) of m*—also depend on the form of the spectrum in the low-frequency region. For the red-noise spectrum, N _eq(N) increases with an increase in N proportionally to N ^1–2α (but not proportionally to N as in the case of bounded spectrum); the correlation scale T ₁(N) is no longer constant (as in the case of bounded spectrum) and increases with an increase in N proportionally to N ^2α.     

Statistical analysis of series of dendrochronological indices

We analyze series of tree-ring indices to understand whether they can be indicators of long-term climate changes. We calculated the synchronous cross-correlation coefficients (CCs) for all possible pairs composed of nine series, of which only one CC (for a pair of the longest series) proved to be sufficiently high (0.72). We revealed a time interval for which this CC is equal to 0.88. We failed to locate sufficiently larger time intervals with a higher CC for other pairs of series. Similar calculations for a series using a low-frequency filter (with allowance for periods of 100 or more years) led to a noticeable decrease in almost all CCs. This confirms the statements circulating in the literature that tree-ring chronologies include climate signals with characteristic times of several decades and that the secular and supersecular oscillations are significantly distorted as a result of standardization and cross-linking. The results presented in this paper show that the complex of series being considered cannot be used as a climate-change indicator without an additional analysis. Here, we describe a possible procedure for such analysis. As a result, we found that only two time series at some time subinterval can provide useful information about climate changes.